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Notes on the Hermitian positive definite solutions of a matrix equation. (English) Zbl 1437.15021

Summary: The nonlinear matrix equation, \(X - \sum_{i = 1}^m A_i^* X^{\delta_i} A_i = Q\), with \(- 1 \leq \delta_i < 0\) is investigated. A fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation is derived. Some properties of the unique Hermitian positive definite solution are obtained. A residual bound of an approximate solution to the equation is evaluated. The theoretical results are illustrated by numerical examples.

MSC:

15A24 Matrix equations and identities
65F45 Numerical methods for matrix equations
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